2.3 – Set Operations and Cartesian Products Intersection of Sets: The intersection of sets A and B is the set of elements common to both A and B. A  B = {x | x  A and x  B} {1, 2, 5, 9, 13}  {2, 4, 6, 9} {2, 9} {a, c, d, g}  {l, m, n, o}  {4, 6, 7, 19, 23}  {7, 8, 19, 20, 23, 24} {7, 19, 23}

2.3 – Set Operations and Cartesian Products Union of Sets: The union of sets A and B is the set of all elements belonging to each set. A  B = {x | x  A or x  B} {1, 2, 5, 9, 13}  {2, 4, 6, 9} {1, 2, 4, 5, 6, 9, 13} {a, c, d, g}  {l, m, n, o} {a, c, d, g, l, m, n, o} {4, 6, 7, 19, 23}  {7, 8, 19, 20, 23, 24} {4, 6, 7, 8, 19, 20, 23, 24}

2.3 – Set Operations and Cartesian Products Find each set. A  B U = {1, 2, 3, 4, 5, 6, 9} A = {1, 2, 3, 4}B = {2, 4, 6}C = {1, 3, 6, 9} {1, 2, 3, 4, 6} {6} {1, 2, 3, 4, 5, 9}  A  BA = {5, 6, 9} B  CC = {2, 4, 5}B = {1, 3, 5, 9)} B  B

2.3 – Set Operations and Cartesian Products Find each set. (A  C)  B U = {1, 2, 3, 4, 5, 6, 9} A = {1, 2, 3, 4}B = {2, 4, 6}C = {1, 3, 6, 9} {2, 4, 5, 6, 9} {5, 9} A = {5, 6, 9} A  C C = {2, 4, 5}B = {1, 3, 5, 9)} {2, 4, 5, 6, 9}  B

2.3 – Set Operations and Cartesian Products Difference of Sets: The difference of sets A and B is the set of all elements belonging set A and not to set B. A – B = {x | x  A and x  B} Note: A – B  B – A {1, 4, 5} {1, 2, 4, 5, 6, } U = {1, 2, 3, 4, 5, 6, 7} A = {1, 2, 3, 4, 5, 6}B = {2, 3, 6}C = {3, 5, 7} A = {7}C = {1, 2, 4, 6}B = {1, 4, 5, 7} Find each set. A – BB – A  (A – B)  C

2.3 – Set Operations and Cartesian Products Ordered Pairs: in the ordered pair (a, b), a is the first component and b is the second component. In general, (a, b)  (b, a) True(3, 4) = (5 – 2, 1 + 3) {3, 4}  {4, 3} False (4, 7) = (7, 4) Determine whether each statement is true or false. False

2.3 – Set Operations and Cartesian Products Cartesian Product of Sets: Given sets A and B, the Cartesian product represents the set of all ordered pairs from the elements of both sets. (1, 6), A = {1, 5, 9} A  B Find each set. A  B = {(a, b) | a  A and b  B} B = {6,7} (1, 7), (5, 6), (5, 7),(9, 6),(9, 7) { } (6, 1), B  A (6, 5), (6, 9), (7, 1),(7, 5),(7, 9) { }

2.3 – Venn Diagrams and Subsets Shading Venn Diagrams: A  B U AB U AB U AB

2.3 – Venn Diagrams and Subsets Shading Venn Diagrams: A  B U AB U AB U A B

2.3 – Venn Diagrams and Subsets Shading Venn Diagrams: A  B U AB U AB U AB A  B in yellow A

2.3 – Venn Diagrams and Subsets Locating Elements in a Venn Diagram Start with A  B U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {2, 3, 4, 5, 6}B = {4, 6, 8} AB U Fill in each subset of U. Fill in remaining elements of U

2.3 – Venn Diagrams and Subsets Shade a Venn diagram for the given statement. (A  B)  C Work with the parentheses. (A  B) AB C U

2.3 – Venn Diagrams and Subsets Shade a Venn diagram for the given statement. (A  B)  C Work with the parentheses. (A  B) B A C U Work with the remaining part of the statement. (A  B)  C

2.3 – Venn Diagrams and Subsets Shade a Venn diagram for the given statement. (A  B)  C Work with the parentheses. (A  B) B A C U Work with the remaining part of the statement. (A  B)  C

2.4 –Surveys and Cardinal Numbers Surveys and Venn Diagrams Financial Aid Survey of a Small College (100 sophomores). 49 received Government grants 55 received Private scholarships 43 received College aid 23 received Gov. grants & Pri. scholar. 18 received Gov. grants & College aid 28 received Pri. scholar. & College aid 8 received funds from all three G C P U 8 (P  C) – (G  P  C) 28 – 8 = (G  C) – (G  P  C) 18 – 8 = (G  P) – (G  P  C) 23 – 8 = – ( ) = – ( ) = – ( ) = – ( ) = 14 14

For any two sets A and B, Cardinal Number Formula for a Region Find n(A) if n(A  B) = 78, n(A  B) = 21, and n(B) = 36. n(A  B) = n(A) + n(B ) – n(A  B) 78 = n(A) + 36 – = n(A) = n(A) 2.4 –Surveys and Cardinal Numbers